We find an analytical solution for the wavefunction that allows us to explicitly depict the generalized Brillouin zone, i.e. the distribution of complex momentum \(\kappa \) , parameterized by \(\beta _{k}\equiv e^{ikL_0}\) , on a complex plane [1]}, [2]}. To this end, we construct a Bloch state for a complex momentum as a traveling wave
\(\hat{\Psi }_{\kappa }=({1}/{\sqrt{N}})\sum _{l=1}^N (\beta _{\kappa })^{l}\hat{b}_l\) ,
obeying, under Hamiltonian (REF ), the equation of motion
\({d \hat{\Psi }_{\kappa }}/{dt} = -i\omega _{\kappa } \hat{\Psi }_{\kappa }-\Gamma _Lg_{\kappa }\hat{\Psi }_{q_*}+\Gamma _Rh_{\kappa }\hat{\Psi }_{-q_*}\)  [3]}, [4]}, [5]}.
The dispersion relation
\(\omega _{\kappa } &=\omega _{\rm K}-i\tilde{\alpha }_G\omega _{\rm K}-i\frac{\Gamma _R}{2}\frac{1+\beta _{\kappa }\beta _{q_*}}{1-\beta _{\kappa }\beta _{q_*}}+i\frac{\Gamma _L}{2}\frac{1+\beta _{\kappa }\beta _{-q_*}}{1-\beta _{\kappa }\beta _{-q_*}}\) 
